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Problems
Contests
National and Regional Contests
Finland Contests
Finnish National High School Mathematics Competition
2002 Finnish National High School Mathematics Competition
2002 Finnish National High School Mathematics Competition
Part of
Finnish National High School Mathematics Competition
Subcontests
(5)
5
1
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On a regular 17-gon
There is a regular
17
17
17
-gon
P
\mathcal{P}
P
and its circumcircle
Y
\mathcal{Y}
Y
on the plane. The vertices of
P
\mathcal{P}
P
are coloured in such a way that
A
,
B
∈
P
A,B \in \mathcal{P}
A
,
B
∈
P
are of diff erent colour, if the shorter arc connecting
A
A
A
and
B
B
B
on
Y
\mathcal{Y}
Y
has
2
k
+
1
2^k+1
2
k
+
1
vertices, for some
k
∈
N
,
k \in \mathbb{N},
k
∈
N
,
including
A
A
A
and
B
.
B.
B
.
What is the least number of colours which suffices?
4
1
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Symmetric convex figure
Convex figure
K
\mathcal{K}
K
has the following property: if one looks at
K
\mathcal{K}
K
from any point of the certain circle
Y
\mathcal{Y}
Y
, then
K
\mathcal{K}
K
is seen in the right angle. Show that the figure is symmetric with respect to the centre of
Y
.
\mathcal{Y.}
Y
.
3
1
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Pairs and probability
n
n
n
pairs are formed from
n
n
n
girls and
n
n
n
boys at random. What is the probability of having at least one pair of girls? For which
n
n
n
the probability is over
0
,
9
?
0,9?
0
,
9
?
2
1
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Classic algebra
Show that if
1
a
+
1
b
+
1
c
=
1
a
+
b
+
c
,
\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{a + b + c},
a
1
+
b
1
+
c
1
=
a
+
b
+
c
1
,
then also
1
a
n
+
1
b
n
+
1
c
n
=
1
a
n
+
b
n
+
c
n
,
\frac{1}{a^n} +\frac{1}{b^n} +\frac{1}{c^n} =\frac{1}{a^n + b^n + c^n},
a
n
1
+
b
n
1
+
c
n
1
=
a
n
+
b
n
+
c
n
1
,
provided
n
n
n
is an odd positive integer.
1
1
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Functional relation
A function
f
f
f
satisfies
f
(
cos
x
)
=
cos
(
17
x
)
f(\cos x) = \cos (17x)
f
(
cos
x
)
=
cos
(
17
x
)
for every real
x
x
x
. Show that
f
(
sin
x
)
=
sin
(
17
x
)
f(\sin x) =\sin (17x)
f
(
sin
x
)
=
sin
(
17
x
)
for every
x
∈
R
.
x \in \mathbb{R}.
x
∈
R
.